Cheapest way from Sapporo to South Korea

flight from sapporo to south korea
flight, train and ferry to south korea
train and ferry to south korea.

I think I want to use a Japan rail pass to see all of japan and travel around hokkaido, but I need to use the sanyo shinkansen to get to the ferry. I can't tell what is a Nozomi train and what isn't.

My visa runs out february 14th, so 15th I'll be in japan illegally. However, I still have internship work on the 13th so I probably need to leave on the 13th in the afternoon.

Buy direct air ticket from Hokkaido.
http://www.new-chitose-airport.jp/en/airport/time/result/?ap=int&type=...

The only ferry service that I know of is that which sails from Hakata (Fukuoka) on the island of Kyushu to Busan. The price is reasonable and the sailing time short., You can surf the internet for info. From Sapporo, you can catch the JR Ltd Express trains to Shin Aomori, connect the JR Shinkansen Hayabusa to Tokyo, than board a JR Shinkansen Hikari to Shin Osaka, and connect the JR Shinkansen Sakura to Hakata. The two types of trains that you cannot cover with the JR Nation Wide Pass is the Nozomi (from Tokyo to Hakata) and Mizuho (from Shin Osaka to Kagoshima Chuo). If you stand on the Shinkansen train platforms, you will be able to see these train names on the train departure notice boards at Tokyo and Shin Osaka. Do not board them. If you are unsure, just get the JR ticketing counter staff to issue reserved seat tickets, they will provide you with the best connections available. Seat reservations are free. If you intend to travel directly from Sapporo to Hakata, you should allocate 2 days travel time, and of course you need to factor in the time you need to catch your ferry, so maybe to be on the safe side 3 days. Alternatively plan to travel from Sapporo , moving gradually southwards so that you will be positioned near Hakata at the end of the trip. I always make allowances to arrive one day earlier before my JR pass expiry in case of earth quakes / heavy snowfall etc that can bring all train services to a halt.

Have you considered Peach airline?
Could be as low as 16000 yens. With 1X transfer in KIX osaka.

http://www.flypeach.com/home.aspx

Train + ferry would be more expensive and too slow to get you out in time.

You could extend your stay by changing your residency status from intern to temporary visitor (tourist) status. Then travel by local train or bus over a few days to catch a ferry.

Without this, fly on one of the low cost airlines. Peach prob your best bet.

If I understand your question, you want to travel your way through Japan as you make your way from Sapporo to South Korea? Or do you plan to go direct without sightseeing along the way?

The OP seemed to suggest that they had just over 24 hours in which to leave the country before becoming an illegal overstayer. So not sure the plans to "see all of Japan" fit into this or if that is a separate idea for the return trip. Cheapest way would still be flying and to accomplish this aim, I suggest booking a window seat.